Now we evaluate the convex roof extension \(\widehat{V}_c(\rho )\) 
analytically. Here we employ the following equality about the
minimum average invariance over all pure state decomposition and the
quantum Fisher information \( F(\rho , A)= \sum _{k,l} 2\frac{(\lambda _k-\lambda _l)^2}{\lambda _k+\lambda _l}\left|{1mu}A_{kl} {1mu}\right|^2\) ,
where \(A_{kl}=\left\langle \phi _k \left| A \right| \phi _l \right\rangle \)  with \(\lbrace \lambda _k,|\phi _k\rangle \rbrace \)  the eigenvalues and the corresponding eigenstates of
the mixed state \(\rho \)  [1]}, [2]}, [3]}, , i.e.,
\(F(\rho , A)=4\min _{\lbrace p_k, |\psi _k\rangle \rbrace } \sum _k p_k\operatorname{Var}(|\psi _k\rangle , A) .\) 
